Multiple solutions for a fractional p-Kirchhoff equation with critical growth and low order perturbations

Abstract

In this article, we deal with the following fractional $ p $-Kirchhoff type equation $ \begin{equation*} \begin{cases} M\left( \int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_p^su=\frac{|u|^{p_\alpha^*-2}u}{|x|^\alpha}+\frac{\lambda}{|x|^\beta} , &\rm \mathrm{in}\ \ \Omega, \\ u>0, &\rm \mathrm{in}\ \ \Omega, \\ u=0, &\rm \mathrm{in}\ \ \mathbb{R}^N\backslash \Omega, \end{cases} \end{equation*} $ where $ \Omega\subset \mathbb{R}^N $ is a smooth bounded domain containing $ 0 $, $ (-\Delta)_p^s $ denotes the fractional $ p $-Laplacian, $ M(t)=a+bt^{k-1} $ for $ t\geq0 $ and $ k>1 $, $ a, b>0 $, $ \lambda>0 $ is a parameter, $ 0<s<1 $, $ 0\leq\alpha<ps<N $, $ \frac{N(p-2)+ps}{p-1}<\beta<\frac{N(p_\alpha^*-1)+\alpha}{p_\alpha^*} $, $ 1<p<pk<p_\alpha^*=\frac{p(N-\alpha)}{N-ps} $ is the fractional critical Hardy-Sobolev exponent. With aid of the variational method and the concentration compactness principle, we prove the existence of two distinct positive solutions.

Keywords

• kirchhoff type equation
• fractional p-laplacian
• critical growth
• variational method
• positive solution